One of the co-founders of SigOpt (YC W15) here. I'm happy to answer and questions about this post or the methods used. More info on the Bayesian methods behind this can be found at sigopt.com/research as well!
We're using Bayesian optimization to tune both the hyperparameters of the unsupervised model and the supervised model, but you are correct that they are being done in unison with the overall accuracy being the target. The lift you get from adding the unsupervised step (and tuning it) is quite substantial (and statistically significant).
The idea of tuning just the unsupervised part (or doing it independently) is great though. All the code for the post is available at https://github.com/sigopt/sigopt-examples/tree/master/unsupe.... It would be interesting to see if doing that would make for a better overall accuracy.
SigOpt isn't a constraint optimization package for solving things like k-SAT thought if that is what you were asking.
You could also try to bake this into the objective function (with an L2 penalty for how bad it violates some constraints), depending on how hard the constraints of the problem actually are.
L2 penalty would likely converge to an infeasible solution. Augmented Lagrangian would be better, but then you're making users handle dual updates. At that point I'd rather use an actual constrained optimization library that does the algorithm carefully, and use primal-dual interior point. Not having this kind of thing built in counts as "no constrained optimization" IMO.